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Normality criteria of meromorphic functions with multiple zeros. (English) Zbl 1175.30030
Let ${\Cal F}$ be a family of meromorphic functions defined in a domain $D\subset\bbfC$. The authors consider a normality criterion for ${\Cal F}$. In this paper, the authors assume a condition of multiple zeros and a certain sharing condition. Let $n$ and $k$ be positive integers $\ge 2$. Suppose that each $f\in{\Cal F}$ has only zeros of multiplicity at least $k$, and suppose that for each pair $(f, g)$ in ${\Cal F}$, $f(f^{(k)})^n$ and $g(g^{(k)})^n$ share a nonzero complex number a ignoring multiplicity. Then it is shown that ${\Cal F}$ is normal in $D$. The main tool of the proof is Zalcman’s lemma.

MSC:
30D35Distribution of values (one complex variable); Nevanlinna theory
30D45Bloch functions, normal functions, normal families
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Full Text: DOI
References:
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